Risk-Sensitive Dividend Problems

We consider a discrete-time version of the popular optimal dividend pay-out problem in risk theory. The novel aspect of our approach is that we allow for a risk averse insurer, i.e., instead of maximising the expected discounted dividends until ruin we maximise the expected utility of discounted dividends until ruin. This task has been proposed as an open problem in H. Gerber and E. Shiu (2004). The model in a continuous-time Brownian motion setting with the exponential utility function has been analysed in P. Grandits, F. Hubalek, W. Schachermayer and M. Zigo (2007). Nevertheless, a complete solution has not been provided. In this work, instead we solve the problem in discrete-time setup for the exponential and the power utility functions and give the structure of optimal history-dependent dividend policies. We make use of certain ideas studied earlier in N. B\"auerle and U. Rieder (2013), where Markov decision processes with general utility functions were treated. Our analysis, however, include new aspects, since the reward functions in this case are not bounded.